Reviving Pascal s and Huygens s Game Theoretic Foundation for Probability. Glenn Shafer, Rutgers University

Transcription

1 Reviving Pascal s and Huygens s Game Theoretic Foundation for Probability Glenn Shafer, Rutgers University Department of Philosophy, University of Utrecht, December 19, 2018 Pascal and Huygens based the calculus of chances on the structure of games, not on frequency. This idea quickly disappeared, because the picture of equally frequent cases was so entrenched. 1

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3 ON THE BACK COVER Ever since Kolmogorov's Grundbegriffe, the standard mathematical treatment of probability theory has been measure theoretic. In this ground breaking work, Shafer and Vovk give a game theoretic foundation instead. While being just as rigorous, the game theoretic approach allows for vast and useful generalizations of classical measure theoretic results, while also giving rise to new, radical ideas for prediction, statistics and mathematical finance without stochastic assumptions. The authors set out their theory in great detail, resulting in what is definitely one of the most important books on the foundations of probability to have appeared in the last few decades. Peter Grünwald, CWI and the University of Leiden 3

4 Two foundations for probability: Measure theory Game theory The game theoretic foundation goes deeper: Probabilities are derived from a perfect information game. To prove a theorem, you construct a strategy in the game. 4

5 Classical picture (Fermat and dice players over the millenia): Equally frequent chances. Modern version (pure mathematics): Probability measure. Space of outcomes. Measure with total measure one. Connect probability with frequency by Bernoulli s theorem (unless event of very small probability happens). Pascal and Huygens and the commercial arithmetics: Players treated equally. Modern version (pure mathematics): Game with three players: Forecaster offers odds. Skeptic decides how to bet. Reality decides the outcomes. Connect probability with frequency by Bernoulli s theorem (unless Skeptic multiplies the capital he risks by large factor). 5

6 1. The calculus of chances before Pascal and Fermat 2. The division problem 3. Pascal s game theory 4. Huygens s game theory 5. Back to frequency 6. Modern game theory 6

7 Laplace launched the legend of Pascal and Fermat in 1795: Probability theory owes its birth to two French geometers of the 17th century. Patriotic words! 7

8 Laplace was more careful in 1814: For quite a long time, people have ascertained the ratios of favorable to unfavorable chances in the simplest games; stakes and gambles were fixed by these ratios. But before Pascal and Fermat, no one gave principles and methods for reducing the matter to calculation, and no one had solved problems of this type that were even a little complicated. So we should attribute to these two great geometers the first elements of the science of probabilities 8

9 Laplace s less nuanced statement is often echoed: Lacroix, 1816: The probability calculus, invented by Pascal and Fermat, has never since ceased exciting the interest and exercising the wisdom of their most illustrious successors Poisson, 1837: A problem about games of chance proposed to an austere Jansenist by a man of the world was the origin of the calculus of probabilities. 9

10 Conceptual revolution? Ian Hacking, born 1936 Ian Hacking, 1976: Probability, as we now conceive it, came into being about It was essentially dual, on the one hand having to do with degrees of belief, on the other, with devices tending to produce stable long run frequencies. Keith Devlin, 2008: The Pascal Fermat correspondence showed that it is possible to use mathematics to see into the future. Keith Devlin, born

11 Before Pascal and Fermat 11

12 Laplace was right: Long before Pascal and Fermat, people ascertained ratios of favorable to unfavorable chances and fixed stakes and bets using these ratios. HOW? 12

13 Long before Pascal and Fermat, people ascertained the ratios of favorable to unfavorable chances, (multiply) fixed stakes and bets using these ratios. (rule of three) 13

14 Latin poem De Vetula, written in Paris about Counted the 216 chances for three dice. Each has the same force and frequency. Sum of points Total # of chances Sum of points Total # of chances David Bellhouse Born 1948 David Bellhouse called De Vetula a medieval bestseller. Nearly 60 copies of manuscript survive. Editors of the printed versions of 1479, 1534, and 1662 understood the counts. 14

15 Long before Pascal and Fermat, people ascertained the ratios of favorable to unfavorable chances, fixed stakes and bets using these ratios. De Vetula counted the chances for the 16 possible sums Sum of points Total # of chances But De Vetula did not turn the counts into probabilities. Probabilities 1/216 3/216 6/216 10/216 15/216 21/216 25/216 27/216 0% 1% 3% 5% 7% 10% 12% 12% Without probabilities, how do you fix stakes and bets? RULE OF THREE 15

16 RULE OF THREE You buy 15 bushels of wheat for 10 shillings. What do you charge for 3 bushels? Answer: 2 shillings. This is the rule of three: find the 4 th number in a proportion from 3 that are known. For us, this is a matter of algebra: 15/3 = 10/x, and so x = 2 shillings. But al-khwarizmi s 9 th -century algebra was all in words. Medieval commercial arithmetics used the rule of three in problem after problem: trading in goods, dividing profits, changing currencies, pricing alloys, etc., etc. Occasionally, for fun, an author might throw in a problem about a game. Algebra with symbols emerged only in the Renaissance, developed by the authors of commercial arithmetics: Italian abbacus masters, German reckoning masters. 16

17 Sum of points Total # of chances Question: Three dice are thrown repeatedly. Player A bets on 9; Player B bets on 15. In other words, Player A gets the money on the table if 9 comes up before 15. Player B gets the money on the table if 15 comes up before 9. Player A puts 5 pistoles on the table. How much should Player B put on the table? Solution: 1. The ratio of 9 s chances to 12 s chances is 25 to So Player A wins only 25 times for every 10 times Player B wins. 3. Player A put 5 pistoles on the table for Player B to win. 4. Player B puts x on the table for Player A to win. 5. So B will win 10 x 5 pistoles every time A wins 25 x x. 6. To make this fair, set x = 2 pistoles: 10 x 5 = 25 x 2. RULE OF THREE 17

18 Scholars who wrote about counting chances before Pascal and Fermat. Geralomo Cardano Galileo Galilei

19 Pascal 19

20 Pascal s research program at the beginning of 1654 included a new field of research: What we call in French faire les partys des jeux, where the uncertainty of fate is so well overcome by the rigor of calculation that each of two players can see themselves assigned exactly what they have coming. Has remained unsettled, because it can only be found by reasoning, not by experience. Treatise will have the surprising title Geometry of Chance. 20

21 Pascal s division problem Pascal s solution 21

22 Pascal s division problem Occasionally discussed in commercial arithmetics. Discussed in print by Pacioli, Cardano and Tartaglio. They gave various solution using the rule of three. Never a game of chance! 22

23 Pascal s two principles: 1. Take any amount you are sure to get. 2. If the game is one of pure chance, and the chances for winning a certain amount are equal, then divide the amount equally. 23

24 Michael Mahoney Fermat s biographer Fermat s solution: There are four equal chances: Player A wins the first round, Player A wins the second round; Player A wins the first round, Player B wins the second round; Player B wins the first round, Player A wins the second round; Player B wins the first round, Player B wins the second round. Player A wins in three our the four chances and so should get ¾ of the stakes. Pascal s response: My solution is better because it carries its demonstration in itself. (Your solution requires experience of frequencies.) 24

25 Pascal s two principles: 1. Take any amount you are sure to get. 2. If the game is one of pure chance, and the chances for winning a certain amount are equal, then divide the amount equally. In the weltlich tradition, always a problem about a ball, archery, or chess tournament. Why assume the game is one of pure chance? In 2003, researchers found a commercial arithmetic in the Vatican library, dating from about 1400, that made Pascal s argument without the pure chance assumption. Anders Hald: The division problem was first solved by Pascal and Fermat in Hald was wrong! 25

26 Huygens 26

27 Pascal s letters to Fermat. Written in Published in Pascal s Triangle arithmétique. Written at the end of Published in Rare. Blaise Pascal Huygens s De Ratiociniis in ludo aleae. Inspired by 1655 visit to Paris. Drafted Published Widely distributed and translated. Christiaan Huygens

28 Having learned about the Pascal Fermat correspondence from mathematicians in Paris, Huygens saw an opportunity to use Descartes s algebra. Use equations to express conditions on a number x. Analysis: Find what x must be if there is a number satisfying the conditions. Synthesis: Prove that the number found does satisfy the conditions. René Descartes

29 Proposition I. If I have the same chance to get a or b it is worth as much to me as (a + b)/2. Consider this fair game: We both stake x. The winner will give a to the loser. The analysis: If I win, I get 2x a. If this is equal to b, then x = (a + b)/2. The synthesis: Having (a + b)/2, I can play with an opponent who stakes the same amount, on the understanding that the winner gives the loser a. This gives me equal chances of getting a or b. Pascal proved the same thing using his two principles and the assumption that the game is one of pure chance. Huygens s argument is better: Clearly does not require that the game be one of pure chance. Replaces principle of equal division by willingness of players to play on even terms. 29

30 Proposition I. If I have the same chance to get a or b it is worth as much to me as (a + b)/2. Consider this fair game: We both stake x. The winner will give a to the loser. The analysis: If I win, I get 2x a. If this is equal to b, then x = (a + b)/2. The synthesis: Having (a + b)/2, I can play with an opponent who stakes the same amount, on the understanding that the winner gives the loser a. This gives me equal chances of getting a or b. Hans Freudenthal Emphasized the difference between Huygens s argument and the modern definition of expectation. 30

31 Proposition III. If I have p chances for a and q chances for b, this is worth (pa + qb)/(p + q). Synthetic (constructive) proof: Assign each chance to a different player. I am one of the p + q players. Each of us puts up (pa + qb)/(p + q). Winner takes all. I make side bet with q opponents; winner gives loser b. I make side bet with other p 1 opponents; winner gives loser a. This gives me p chances for a and q chances for b. (p + q) (pa + qb)/(p + q) qb (p 1)a = a 31

32 Proposition III. If I have p chances for a and q chances for b, this is worth (pa + qb)/(p + q). Synthetic (constructive) proof: Assign each chance to a different player. I am one of the p + q players. Each of us puts up (pa + qb)/(p + q). Winner takes all. I make side bet with q opponents; winner gives loser b. I make side bet with other p 1 opponents; winner gives loser a. This gives me p chances for a and q chances for b. (p + q) (pa + qb)/(p + q) qb (p 1)a = a Ivo Schneider Born 1938 Ivo s objection to Huygens: Not all players are treated the same if one gets to decide what side bets to make. 32

33 At the end of De rationciniis, Huygens stated 5 problems with answers but no solutions. Fermat had proposed Problems 1 and 3. Pascal had proposed Problem 5. We can justify Laplace s words with the claim that probability theory was launched by these problems. Huygens s 5 th problem: Having each taken 12 coins, A and B play with 3 dice. A gives a coin to B each time he gets an 11. B gives a coin to A each time he gets a 14. The winner be the one who first has all the coins. The ratio of A s chance to B s chance is to

34 The first surviving document showing with an event tree is in a manuscript dated August 1676, in which Huygens solves Problem 5. 34

35 Back to Frequency 35

36 Pascal s and Huygens s game-theoretic foundations were quickly pushed aside by the deeply entrenched concept of equally frequent chances. Pierre Rémond de Monmort Essay d analyse sur les jeux de hazard 1708 Abraham De Moivre De mensura sortis 1711 Jacob Bernoulli Ars conjectandi

37 Conclusion 37

38 Pascal and Fermat provided a game theoretic foundation for probability theory. But it did not satisfy the standards of rigor now demanded by game theory. What are the rules of play? As Ivo Schneider asks, why does one player get to make the side bets he wants? 38

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40 Fermat and the dice players: Equally frequent chances. Make it modern, pure mathematics: Start with probability measure. Space of outcomes. Measure with total measure one. Obtain frequencies by Bernoulli s theorem (unless event of very small probability happens). Pascal and Huygens and the commercial arithmetics: Players treated equally. Make it modern, pure mathematics: Start with game with three players: Forecaster offers odds. Skeptic decides how to bet. Reality decides the outcomes. Obtain frequencies by Bernoulli s theorem (unless Skeptic multiplies the capital he risks by large factor). 40